# A Computational Tropical Geometry Framework for Neural Networks

arXiv: https://arxiv.org/abs/2405.20174 · HTML: https://mlsanity.com/paper/2405.20174/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 39 references that are in the map.
- First posted on arXiv 30 May 2024.

## Builds on (most cited first, 24 of 24)

- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [Benefits of depth in neural networks](https://mlsanity.com/paper/1602.04485/) (2016)
- [Bounding and Counting Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1711.02114/) (2017)
- [Deep ReLU Networks Have Surprisingly Few Activation Patterns](https://mlsanity.com/paper/1906.00904/) (2019)
- [Tropical Geometry of Deep Neural Networks](https://mlsanity.com/paper/1805.07091/) (2018)
- [On the number of response regions of deep feed forward networks with piece-wise linear activations](https://mlsanity.com/paper/1312.6098/) (2013)
- [Parallelizing the dual revised simplex method](https://mlsanity.com/paper/1503.01889/) (2015)
- [Model Complexity of Deep Learning: A Survey](https://mlsanity.com/paper/2103.05127/) (2021)
- [On the Number of Linear Regions of Convolutional Neural Networks](https://mlsanity.com/paper/2006.00978/) (2020)
- [When Deep Learning Meets Polyhedral Theory: A Survey](https://mlsanity.com/paper/2305.00241/) (2023)
- [Sharp bounds for the number of regions of maxout networks and vertices of Minkowski sums](https://mlsanity.com/paper/2104.08135/) (2021)
- [Empirical Studies on the Properties of Linear Regions in Deep Neural Networks](https://mlsanity.com/paper/2001.01072/) (2020)
- [SplineCam: Exact Visualization and Characterization of Deep Network Geometry and Decision Boundaries](https://mlsanity.com/paper/2302.12828/) (2023)
- [A Tropical Approach to Neural Networks with Piecewise Linear Activations](https://mlsanity.com/paper/1805.08749/) (2018)
- [The Real Tropical Geometry of Neural Networks](https://mlsanity.com/paper/2403.11871/) (2024)
- [Empirical Bounds on Linear Regions of Deep Rectifier Networks](https://mlsanity.com/paper/1810.03370/) (2018)
- [Algorithmic Determination of the Combinatorial Structure of the Linear Regions of ReLU Neural Networks](https://mlsanity.com/paper/2207.07696/) (2022)
- [On the Number of Regions of Piecewise Linear Neural Networks](https://mlsanity.com/paper/2206.08615/) (2022)
- [Exact Count of Boundary Pieces of ReLU Classifiers: Towards the Proper Complexity Measure for Classification](https://mlsanity.com/paper/2306.08805/) (2023)
- [ReLU Neural Networks, Polyhedral Decompositions, and Persistent Homolog](https://mlsanity.com/paper/2306.17418/) (2023)
- [An easily computable upper bound on the Hoffman constant for homogeneous inequality systems](https://mlsanity.com/paper/2302.02193/) (2023)
- [An algorithm to compute the Hoffman constant of a system of linear constraints](https://mlsanity.com/paper/1804.08418/) (2018)

## Built on by (most cited first, 1 of 1)

- [The Computational Complexity of Counting Linear Regions in ReLU Neural Networks](https://mlsanity.com/paper/2505.16716/) (2025)

## Abstract (arXiv)

We propose a computational tropical geometry framework for the symbolic analysis of neural networks with tropical activations. The number of linear regions of a neural network has been actively studied as a measure of the expressivity of a given architecture. To study these, we work in the setting of tropical geometry---a combinatorial and polyhedral variant of algebraic geometry---where there are known connections between tropical rational maps and feedforward neural networks. We expand this connection by developing concrete computational tools for studying the linear regions of neural networks. We present an algorithm, together with a proof of correctness, which computes the linear regions of a neural network as explicit unions of polyhedra. We further relate the computation of the number of linear regions of a tropical expression to the number of monomials that appear in it, and show how tropical expressions can often be pruned to remove redundant monomials. We introduce the Hoffman constant of a neural network's tropical expression, a geometric quantity that controls the distance from any point in the input space to the farthest linear region. We provide the open source Julia library TropicalNN.jl, which is built on top of the OSCAR computer algebra system and implements the algorithms mentioned above to analyze neural networks symbolically using their tropical representations. We present a set of proof-of-concept computational examples to demonstrate how our tropical geometric theory can be applied to reveal insights on the expressivity of a network architecture.